QuestionIntegrate: $\int \frac{2 x+5}{\sqrt{4 x^{2}-20 x+29}} d x$ (15 pts, 1a: 5 pts)

Studdy Solution

STEP 1

Assumptions1. We are given the integral $\int \frac{ x+5}{\sqrt{4 x^{}-20 x+29}} d x$ . We are required to find the integral of the given function

STEP 2

The given integral is a complex one. It is not immediately obvious how to integrate it. However, it looks like a derivative of a function in the form of $u'/\sqrt{u}$ , where $u$ is a quadratic function. So, we can try to express the numerator in the form of the derivative of the denominator.

STEP 3

The denominator is a square root of a quadratic function. Let's denote it as $u$ and find its derivative.
$u =x^2 -20x +29$ $u' =8x -20$

STEP 4

We can see that the derivative of $u$ is similar to the numerator of the integral, but not exactly the same. However, we can express the numerator as a sum of the derivative of $u$ and a constant.
$2x + = (8x -20) /4 +$

STEP 5

Now, we can rewrite the integral as follows $\int \frac{2 x+5}{\sqrt{4 x^{2}-20 x+29}} d x = \frac{1}{4} \int \frac{u'}{\sqrt{u}} dx +5 \int \frac{1}{\sqrt{u}} dx$

STEP 6

The first integral on the right side can be integrated using the formula for the integral of $u'/\sqrt{u}$ , which is $2\sqrt{u}$ . The second integral is a standard integral of the form $1/\sqrt{u}$ , which is $2\sqrt{u}$ .
$\frac{1}{4} \int \frac{u'}{\sqrt{u}} dx +5 \int \frac{1}{\sqrt{u}} dx = \frac{1}{4} \cdot2\sqrt{u} +5 \cdot2\sqrt{u}$

STEP 7

Substitute $u$ back into the equation.
$\frac{1}{4} \cdot2\sqrt{4x^2 -20x +29} +5 \cdot2\sqrt{4x^2 -20x +29}$

STEP 8

implify the equation.
$\frac{1}{2}\sqrt{4x^2 -20x +29} +10\sqrt{4x^2 -20x +29}$

STEP 9

Combine like terms.
$\frac{21}{2}\sqrt{4x^2 -20x +29}$ So, the integral of the given function is $\frac{21}{2}\sqrt{4x^2 -20x +29} + C$ , where $C$ is the constant of integration.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms